CHEMICAL POTENTIAL• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students. • I plan to go through these slides in one 90-minute lecture.

Zhigang Suo, Harvard University

The play of thermodynamics

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energy space matter charge

ENTROPY

temperature pressure chemical potential electrical potential

heat capacity compressibility capacitanceHelmholtz function enthalpy

Gibbs functionthermal expansionJoule-Thomson coefficient

plan

• Definition of chemical potential• Examples of chemical potential• Equilibrium of two systems• Equilibrium of a chemical reaction• Equilibrium of phases

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Model an open system as a family of isolated systems

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liquid

2Ogas a family of isolated systemsof four independent variables:U, V, NH2O, NN2

• The wine contains many species of molecules (components) and two phases.• The wine is an open system, exchanging energy, space, and two components

with the rest of the world.• Make the wine an isolated system by insulating the bottle, jam the piston, and shut

the valves. • A system isolated for a long time reaches a state of thermodynamic equilibrium.• Define the entropy of the isolated system: S = log (number of quantum states).• Isolating the wine at various values of (U,V, NH2O, NN2), we obtain a family of

isolated systems of four independent variables.• Model the family of isolated systems by function S(U,V, NH2O, NN2).

liquid

weights

fire

gas

valve for N2

open system

valve for H2O H2O tank

N2 tank

Derivative1. an operation in calculus

2. a thing based on something else

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liquid

2Ogas

a family of isolated systemsS(U, V, NA, NB)

Name derivatives by Gibbs equations

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Define temperature:

Define pressure:

Define chemical potential:

Define chemical potential

Calculus:

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• These equations define the two chemical potentials.• Each chemical potential is a child of entropy and a component.• Chemical potential of a component is an intensive property of a system.• T appears in the definition by convention.• Negative sign appears in the definition by convention. Thus, an isolated system increases

entropy when a component goes from a place of high chemical potential to a place of low chemical potential.

• Grammar: The chemical potential of a component in a system (e.g., A is the chemical potential of water in the wine, and B is the chemical potential of nitrogen in the wine).

Notes on chemical potentials

Why don’t we know chemical potential as well as temperature?

• Blame our parapets. Our parents tell us a lot about temperature, but never tell us about chemical potential. (But they do tell us about humidity, and smells of many kinds.)

• Blame our world. The world confuses us with many species of molecules.

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Breed equations (Gibbs 1878)

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Gibbs equation:

Solve for dU:

Calculus:

Breed more equations

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Gibbs equation:

A Legendre transform defines the Gibbs function:

Combine the above two equations:

Calculus:

On being extensive

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Gibbs equationAll independent variables are extensive:

Increase all extensive properties proportionally:

Derivative with respect to :

Solve for U:

Definition of the Gibbs function:

Combine the above two equations:

plan

• Definition of chemical potential• Examples of chemical potential• Equilibrium of two systems• Equilibrium of a chemical reaction• Equilibrium of phases

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Pure substance

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Define Gibbs function per molecule (or per mole):

Define chemical potential of a pure substance:

Compare the two definitions:

Recall the definition of the Gibbs function:

Recall the Gibbs equation:

1. For a pure substance, we know how to measure TVPUS.2. From TVPUS we can calculate the chemical potential .3. Chemical potential requires the absolute entropy.

Incompressible pure substance

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Gibbs equation:

Incompressibility:

Integration:

Ideal gas

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Gibbs equation:

Law of Ideal gas:

Integration:

Look up values:

P0 = 1 atm

Ideal-gas mixture

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partial pressure of component A

chemical potential of component A in an ideal-gas mixture

chemical potential of pure ideal gas A at 1 atm

The chemical potential of a component in an ideal-gas mixture is the same as the chemical potential of the component in the pure gas, provided we use the partial pressure of the component.

Chemical potential of water in moist air relates to relative humidity

Model the moist air as an ideal-gas mixture

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plan

• Definition of chemical potential• Examples of chemical potential• Equilibrium of two systems• Equilibrium of a chemical reaction• Equilibrium of phases

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Two systems exchanging energy, space and molecules

Isolated system conserves energy, space, and matter over time:dU’ + dU’’ = 0. dV’ + dV’’ = 0dNA’ + dNA’’ = 0dNB’ + dNB’’ = 0

Isolated system not in equilibrium generates entropy over time: dS’ + dS’’ > 0Isolated system in equilibrium keeps entropy constant over time: dS’ + dS’’ = 0

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U’, V’, NA’, NB’S’(U’, V’, NA’, NB’)

isolated systemU’’, V’’, NA’’, NB’’S’’(U’’, V’’, NA’’, NB’’)

diathermal, moving, permeable to components A and B

open system (‘) open system (‘’)

Equilibrium of two systems

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U’, V’, NA’, NB’S’(U’, V’, NA’, NB’)

isolated systemU’’, V’’, NA’’, NB’’S’’(U’’, V’’, NA’’, NB’’)

diathermal, moving, permeable to components A and B

open system (‘) open system (‘’)

Thermal equilibrium:

Mechanical equilibrium:

Chemical equilibrium of component A:

Chemical equilibrium of component B:

Measuring chemical potential of a component in a system

chemical potential of water in the wineChemical potential affects everything. Everything measures chemical potential

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liquid

weights

fire

gasopen system

A membrane permeable to H2O only

pure H2O(T,P)

Hygrometerhumidity sensors

First inventor: Johann Heinrich Lambert (1755)Humidity affects everything. Everything is a hygrometer.

Today’s opportunity: The Internet of things.

22https://en.wikipedia.org/wiki/Hygrometer

• Bimaterial strip• Hair-tension hygrometer• Wet-bulb and dry-bulb• Dew-point hygrometer• Capacitor• Resistor• Thermal conductivity• Weight

Sensors for chemical potentials of various components

• Humidity sensor• pH sensor• Oxygen sensor• CO2 sensor

• Electronic nose

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Henry’s law (1803)

N2 dissolved in wateryN2

N2 in airPN2

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Solubility

N2 dissolved in rubbermole/volume

N2 in airPN2

plan

• Definition of chemical potential• Examples of chemical potential• Equilibrium of two systems• Equilibrium of a chemical reaction• Equilibrium of phases

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weights

Reaction

thermal reservoir, T

piston

Fix T,PChange U,VNA, NB, NC, ND,

reactionchamber

A chemical reaction conserves the number of atoms in each species

(i are stoichiometric coefficients):

Increment of the number of each component ( is the degree of reaction):

Conservation of energy:

Entropy is additive:

Isolated system (IS)

Qout

Equilibrium of a reaction

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Conservation of atoms:

Conservation of energy:

Entropy is additive:

Condition of equilibrium:

Calculus and definitions:

Chemical equilibrium:

Ideal-gas reaction

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Chemical reaction:

Condition of equilibrium:

Chemical potential of a component in an ideal- gas mixture:

Define

Condition of equilibrium:

Equilibrium constant:

van’t Hoff equation

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Equilibrium constant:

Algebra:

Calculus:

Recall:

Recall:

van’t Hoff equation:

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Simultaneous reactions

Reaction 1

Reaction 2

plan

• Definition of chemical potential• Examples of chemical potential• Equilibrium of two systems• Equilibrium of a chemical reaction• Equilibrium of phases

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Equilibrium of two phases

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U’, V’, NA’, NB’S’(U’, V’, NA’, NB’)

isolated systemU’’, V’’, NA’’, NB’’S’’(U’’, V’’, NA’’, NB’’)

phase (‘) phase (‘’)

Thermal equilibrium:

Mechanical equilibrium:

Chemical equilibrium of component A:

Chemical equilibrium of component B:

In equilibrium, the two phases have the same temperature, the same pressure, and the same chemical potential of each component. A total of 2 + C equations. C = number of components.

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IV = 2 + C - PH

IV = number of independent variablesC = number of componentsPH = number of phases in equilibrium

The Gibbs phase rule

C components: 1, 2,…, CPH phases: (‘), (‘’),…Composition of phase (‘): y’1, y’2,…y’C-1

Composition of phase (‘’): y’1, y’2,…y’C-1

…All phases have the same T and the same P. 2 variablesTotal number of number fractions: PH(C-1)Chemical potential of each component is the same in all phases: (PH-1)C equationsIV = 2 + PH(C-1) –(PH-1)C

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Gibbs rule: IV = 3 - PH

IV = number of independent variablesPH = number of phases in equilibrium.

The Gibbs phase rulePure substance, C = 1

Number of phases in equailibrium PH IV

Single phase 1 2

Two phases in equilibrium (two-phase boundary) 2 1

Three phases in equilibrium (triple point) 3 0

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Gibbs rule: IV + PH = 4

IV = number of independent variablesPH = number of phases in equilibrium.

Two-component (binary) system, C = 2

Set pressure at a fixed pressureA diagram of two variables, T and yB

Three phases in equilibrium in a binary mixture: eutectic point

37https://en.wikipedia.org/wiki/Eutectic_system

Water-salt phase diagram

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Gibbs rule: IV + PH = 5

IV = number of independent variablesPH = number of phases in equilibrium.

Three-component (tertiary) system, C = 3Set pressure at a fixed valueSet temperature at a fixed value

Gibbs triangle: Each point in the triangle represents a composition, , yB and yC

Stainless steel phase diagram at 900 degrees Celsius (ASM 1-27)

Summary• Chemical potential is a child of entropy and a

component.• Chemical potential (of a species of molecules) in

a pure substance coincides with the Gibbs function per molecule.

• Chemical potential of a component in an ideal-gas mixture is the same as that of the pure component, provided we use partial pressure of the component.

• Use chemical potential to analyze equilibrium of systems, reactions, and phases.

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